In a variety of applications where nanoscopic particles are involved, such as for example in the field of nanomedicines, virology, nanotoxicology, biosafety, etc., determination of particle concentration in a dispersion is required. Different techniques for determining particle concentration are known.
Some techniques for determining particle concentration in a dispersion make use of microfluidic absorption measurements or fluorescence measurements. The concentration is then determined based on the absorbance properties or fluorescent properties of particles or labels bound to the particles. Another technique used for determining particle concentration is the use of assays. Typically, particles in a dispersion are then specifically bound to labels and the concentration is determined based on the number of labels that can be measured.
Another example for determining particle concentration, whereby no binding of labels to the particles of interest is required, is the use of a Coulter Counter® whereby particles are allowed to flow through a microcapillary of suitable size such that only one particle at a time can reach the measurement region. Particles are detected based on a change in electrical impedance in the liquid filled microchannel. In order to be able to measure such particles, the particles should be larger than 0.4 μm and should have a low electrical conductance.
Analysis of trajectories of dynamic particles using optical microscopy provides a powerful approach to both characterizing particle motion as well as estimating stochastic motion-related parameters, e.g. diffusion coefficient or particle size, and distributions of motion related parameters. Single particle tracking (SPT) is increasingly used as the method of choice to study the behavior of complex systems at small spatial and temporal scales, where traditional ensemble-averaging methods are unable to give a satisfactory account of the complexity of driving processes and function. With the development of tools unhindered in capability by the inherent smoothing effect due to averaging over many particles or over long time lapses, a wealth of information previously beyond reach is now accessible in everyday practice. It is indeed likely that different approaches to the tracking of individual particles or even individual molecules will revolutionize the biophysical and pharmaceutical measurement techniques as these methods become mature and spread. This progress promises to elucidate many aspects of the fundamental interaction of nanomaterials with biological media in the context of nanomedicine, biomedical imaging, medical diagnostics and nanotoxicology. In practice, using image processing, the motion trajectories can be obtained for individual particles that are visible in the SPT movies. Each trajectory can then be analyzed to obtain information on the movement of a particular particle. For example, when particles are undergoing Brownian diffusion, which is e.g. the case for submicron particles in a viscous dispersion, one can estimate the diffusion coefficient of each particle. Since the diffusion coefficient is inversely proportional to the size of the particle, it is possible to perform accurate size measurements of dispersed particles by SPT. Determination of particle concentration in the setting of single particle tracking sometimes is performed by counting the number of particles in an image or the number of particle trajectories that are observed in a given time interval, as e.g. described in Malloy A. and Carr B., Part. Part. Syst. Charact. 23 (2006) 197-204. Converting the observed number of trajectories into a number concentration requires accurate knowledge of the effective volume in which the particles are observed by the SPT instrument. The size of the effective observation volume is typically assumed to be known a priori when the aim is to compute or estimate other physical parameters such as concentration or diffusion coefficient, see for example N. H. Bingham and B. Dunham, “Estimating diffusion coefficients from count data: Einstein-Smoluchowski theory revisited”, Ann. Inst. Statist. Math, 49(1997), 667-679. However, it was noted by Malloy and Carr that the effective observation volume depends on the size of the particles and intensity of the light emitted or scattered by the particles. Furthermore, it is clear to a person skilled in the art that the effective observation volume also depends on the signal to noise ratio, background intensity and the image processing settings that are used to detect the particles in the SPT movies. Accurate calibration of the effective detection volume is, therefore, generally not possible by means of a reference measurement using reference particles of known concentration. This is because the actual particles under study and the medium in which they are dispersed might be different from the reference set. Also, the image processing settings may be very different in both cases leading to an altered effective observation volume.
Another shortcoming of SPT concentration measurements based on the number of particle trajectories that was recognized by Malloy and Carr, is that fast moving particles (e.g. small particles) can enter and leave the observation volume more frequently than slow moving particles (e.g. large particles) in the same time interval. Without a suitable correction this will lead to a biased concentration measurement (sampling bias) in that the number of fast moving particles are overestimated compared to more slowly moving particles.